loading forces according to [14]. Simultaneously, the technical mechanic distinguishes between static and kinetic systems. In doing so, the kind of system depends on its tech-nical constraints and thus, on its degrees of freedom. The multitude of mechatech-nically regarded systems are static systems, like the studied adhesively bonded composite parts in this work.
In general, the mechanical structure analysis is a useful tool to dimension whatever kind of structural component. By calculating the deformation and strains of a component under specic load cases, its stress-tensor can be determined. Furthermore, the existing analyzed stresses can afterwards be set into relation with the stress allowable of the ma-terial. This gives information whether the structure fails or resists the applied loads.
A proper structure analysis is done to guarantee necessary safety against the failure of a structural component. The structure analysis goes always side by side with mechan-ical tests proving the suitable eld of application for the analysis. In economic regards, the application of a structure analysis has its main purpose in saving cost expended for the materials, the assembly and the tness of a component.
On the one hand, the structure analysis benets fewer necessary tests (s. [29]). On the other hand, a simultaneous process of test and analysis (as you can see in gure 1.2) leads to more know-how in the development of technical components.
A.2.1. Linear Analysis
The linear static analysis is the most commonly used form of structure analysis. It is based on the linear stiness K of a component [16]. The stiness is a property which describes the deformation u of a loaded component by an applied load P .
P = K u (24)
In this connection the stiness K is dependent on dierent factors.
Primarily, the material of a component inuences its stiness. A linear material behavior is expressed by the elasticity law, which has been established in 1676 by R. Hooke. The elasticity law relates strains and stresses in a material. The dependent coecient
be-to deform plastically and hence, the modulus of elasticity decreases.
Secondly, the geometrical shape of a component codetermines its stiness. If the de-formation of a structural component is high or takes inuence on the loading behavior, geometrical nonlinearity is present. A rule of thumb states according to [16] that geo-metrical nonlinearities have to be respected if a deformation of more than a twentieth part of the largest component dimensioning is present.
The last inuence on the stiness is given by the boundary conditions of a component.
If a dierent clamping of a component is chosen it eects the deformation at an un-changed load.
If none of these three factors inuence the stiness signicantly, a linear analysis of the component is appropriate. A linear stiness leads to the fact that the analyzed compo-nent will gain back its initial state when its loading forces P are removed.
The linearity between loading and deformation oers a simple principle of superposi-tion of single loads to get the results of a complex load case of multiple loads.
A.2.2. Nonlinear Analysis
In contrast to the linear structure analysis, the nonlinear structure analysis is designed to analyse a structure in which signicant stiness changes are present. Material, ge-ometrical or boundary nonlinearities [1] lead, thus, to the following change of equation (24):
P = K(P; u) u (25)
The changing of the stiness matrix K accounts for an iterative solution algorithm to get accurate results. The quantity of nonlinearities in the stiness matrix determines appropriate parameters for the solution algorithm. If a high-grade area of nonlinearities is reached in a structural analysis, this area needs to be more accurately simulated. The convergence of the constitutive equation obtains in this context an important meaning.
Convergence
To satisfy the nonlinear stiness changing in the model, the nal state of deformation
value is reached. The breakdown of the analysis into several increments implies that the computing time rises consequently with the amount of increments.
The aim for a suitable FEM model is always to get most eciently accurate results.
Thus, the computing costs should stay at a low level, while the analysis results should reach a certain level of quality. The level of quality is expressed by the convergence fault of the analysis.
Figure A.4: Visualisation of an explicit method of resolution
In gure A.4, a comparison between the analytical correct solution (green curve) and the solution of an explicit nonlinear solution algorithm (red curve) is depicted.
Thus, the aim of the nonlinear numerical analysis is in this case to accurately reproduce the analytical function. In this purpose, the nonlinear analysis is divided into the typi-cal increments. At each increment in the explicit nonlinear analysis the typi-calculated end function value of the previous increment is taken as the start function value for the next increment. In order to calculate the end function value of an increment the function is derived at the start point. Using the derivation at the start point, the gradient in this point is determined so that a linear tangent approach can be fullled for the approxi-mation of the increment. The higher the curvature of the function and the taller the chosen increment, the heavier the convergence fault ERRconv;explicit rises. The latter is demonstrated in gure A.4, by comparing the left and the right image.
The explicit solution algorithm provides a relatively fast approach for a nonlinear analy-sis. Unfortunately, it leads to a summation of convergence faults of each increment and
for instance to simulate crashes or impacts.
In order to control the quantity of the convergence faults the implicit solution algorithm is introduced. In each increment, the solver tries to satisfy the algebraic transformed equation (24).
Panalytisch kanalytisch uanalytisch = 0 (26)
The implicit solution algorithm shows its benets on static, quasi-static and long duration events. In the analyzing process, multiple iterations are solved during one single increment until the convergence fault ERRconv;implicitof the current increment falls below a specied value, as it is mapped in gure A.5.
Figure A.5: Visualisation of an implicit method of resolution
A popular method to numerically evaluate a function value at a certain time step of a specic function is the Newton-Raphson-Method. This method is often used to iteratively approximate an analytical function.
Pimplicit;i kimplicit;i uimplicit;i = Errorimplicit;i 0 (27) For the present work, the nonlinear results are approximated along the recommendation of [5] by using the implicit solution algorithm and the Newton-Raphson-Method.
There exist three main methods in a nonlinear structural FEM-analysis to control the convergence using the implicit solution algorithm (s. [1]). These three methods result out of the parameters of the constitutive equation (27).
under a specied value, convergence is achieved for the increment.
The second method of convergence controlling is to compare the maximum displace-ments (U) of the current iteration with the displacedisplace-ments of the previous iteration. The dierence between both has to get under a certain level.
The third method is the control of strain energy (W) which basically works like the second method. By this method the whole model is checked iteratively.
MSC Nastran proposes to use a convergence error of 0,1 to mechanically analyze a struc-ture. This is kept within all the executed analysis. The convergence check is done via the force and the displacement convergence method (UPV).
Because the discretisation is based on iterative linear approximations of the constitutive equation, the implicit solution algorithm needs more iterations to obtain convergence, in areas, where the constitutive equations have high nonlinearities.
material parameter
Figure B.1: shear stress shear distribution over adhesive layer, for linear-elastic un-damaged behavior at an applied displacement in x-direction of 0,0078mm
Figure B.2: shear stress shear distribution over adhesive layer, for plastic behavior, with partly damaged CZE at an applied displacement in x-direction of 0,0118mm
En 4035MPa (LTSM, Uni Patras) [23]
G 720MPa (LTSM, Uni Patras) BBHC tests [23]
n 64MPa (LTSM, Uni Patras) [23]
shear 35MPa (LTSM, Uni Patras) BBHC tests [23]
GCI 0,32N/mm (LTSM, Uni Patras) average of area
calc. and usage of SBT [23]
GCII 0,5N/mm (LTSM, Uni Patras) average of CBBM
and Norm:AITM [23]
1 0,546875 tsheartn
2 1,52439 GshearGn
1
Kel;n 4035N=mm2
thCZE equation 12 Kel;shear 720N=mmthCZE 2 equation 12 c;n 0; 01586 thCZE = KCZEntn c;shear 0; 04861 thCZE = KCZEsheartshear
max;n 0,01025mm = Gtnn2
max;shear 0,02857mm = Gtshearshear2
Table 8: Material parameter of Loctite Hysol EA9695-adhesive
Parameter magnitude origin
En Rohrprobenprogramm DLR [31]
G Loctite Report [41]
n equation 20
shear Loctite Report [41]
GCI 1,01852N/mm Tinius Olsen H5K-S UTM tensile test GCII 0,78341N/mm [35]MTS universal test [35]
1 tsheartn
2 Gshear
Gn
1
Kel;n equation 12
Kel;shear equation 12
c;n = KCZEntn
c;shear = KCZEsheartshear
max;n = Gtnn2
max;shear = Gtshearshear2
Table 10: Material-MCOHE-input for the CZEs simulating the Mojo-Mix-adhesive according to [2] and table 7
MCOHE MID 1 +
+ 0.328 9.517e-5 0.01025 0.546875 +
+ 1.52439
Table 11: Material-MCOHE-input for the CZEs simulating the Loctite Hysol EA9695-adhesive according to [2] and table 8
MCOHE MID 1 +
+ 1.01852 +
+ 0.769165
Table 12: CFC Material Hexcel IM7 8552 [30] (p.1)
terial input has been executed at the CLS-model.
Primarily, the change of parameters is done to see, whether a change of the material input can raise the failure load of the CLS-component from Fsim;crit=30kN to the level of Ftest;crit=55kN.
In a rst step of parameter variation, the critical energy release rates GCI and GCII
stay unmodied (values according to [35]). The shear-normal ratios 1 and 2 remain x as well. The initial CZE-stiness Kel and the critical traction max of the CZE have been varied as can be seen on gure C.1. The red colored curve represents the original bilinear material law for normal loading direction (s. gure 3.3) as it has been utilized at the presented CLS-model.
As 1 and 2 remain unchanged the material-law for pure shear loading changes simul-taneously to the modication of the material-law for pure normal loading.
In a second step the same parameter modications have been chosen, with a changed energy release rate (s. gure C.2).
In order to summarize the results of the executed parameter study, the failure loads of the CLS-model have been plotted next to their corresponding material curves of the CZEs.
Figure C.1: Variation of CZE-parameter
with GCI=0,328N/mm Figure C.2: Variation of CZE-parameter with GCI=0,492N/mm
The result of the parameter study proves, that a change of cohesive stiness has just small inuence on the failure load. This is expected, because the CZE-stiness has
critical failure load Fsim;crit. If an increase of the failure load is desired, the changing of the energy release rate is the recommended parameter. On the one hand, it comparably results in a larger eect and on the other hand, the changing of the critical traction can be used to compensate the disturbance of mesh size eects pursuant [25].
Additionally the ESA states in [9], that the shear strength of adhesively bonded struc-tural joints can be better expressed by the strain energy to failure per unit bond area, than by any of the individual properties such as peak shear stress.